Properties

Label 2-177-1.1-c9-0-5
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.23·2-s + 81·3-s − 510.·4-s − 1.72e3·5-s + 99.9·6-s + 4.57e3·7-s − 1.26e3·8-s + 6.56e3·9-s − 2.12e3·10-s − 7.56e4·11-s − 4.13e4·12-s − 1.28e5·13-s + 5.65e3·14-s − 1.39e5·15-s + 2.59e5·16-s − 7.94e3·17-s + 8.09e3·18-s − 2.91e5·19-s + 8.79e5·20-s + 3.70e5·21-s − 9.33e4·22-s − 5.86e5·23-s − 1.02e5·24-s + 1.01e6·25-s − 1.59e5·26-s + 5.31e5·27-s − 2.33e6·28-s + ⋯
L(s)  = 1  + 0.0545·2-s + 0.577·3-s − 0.997·4-s − 1.23·5-s + 0.0314·6-s + 0.720·7-s − 0.108·8-s + 0.333·9-s − 0.0672·10-s − 1.55·11-s − 0.575·12-s − 1.25·13-s + 0.0393·14-s − 0.711·15-s + 0.991·16-s − 0.0230·17-s + 0.0181·18-s − 0.513·19-s + 1.22·20-s + 0.416·21-s − 0.0849·22-s − 0.436·23-s − 0.0628·24-s + 0.518·25-s − 0.0682·26-s + 0.192·27-s − 0.718·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.6100569661\)
\(L(\frac12)\) \(\approx\) \(0.6100569661\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81T \)
59 \( 1 + 1.21e7T \)
good2 \( 1 - 1.23T + 512T^{2} \)
5 \( 1 + 1.72e3T + 1.95e6T^{2} \)
7 \( 1 - 4.57e3T + 4.03e7T^{2} \)
11 \( 1 + 7.56e4T + 2.35e9T^{2} \)
13 \( 1 + 1.28e5T + 1.06e10T^{2} \)
17 \( 1 + 7.94e3T + 1.18e11T^{2} \)
19 \( 1 + 2.91e5T + 3.22e11T^{2} \)
23 \( 1 + 5.86e5T + 1.80e12T^{2} \)
29 \( 1 + 4.07e6T + 1.45e13T^{2} \)
31 \( 1 + 2.85e6T + 2.64e13T^{2} \)
37 \( 1 + 1.13e7T + 1.29e14T^{2} \)
41 \( 1 - 2.10e7T + 3.27e14T^{2} \)
43 \( 1 + 1.69e7T + 5.02e14T^{2} \)
47 \( 1 - 2.55e7T + 1.11e15T^{2} \)
53 \( 1 - 4.62e7T + 3.29e15T^{2} \)
61 \( 1 - 5.77e5T + 1.16e16T^{2} \)
67 \( 1 + 1.20e8T + 2.72e16T^{2} \)
71 \( 1 - 2.52e8T + 4.58e16T^{2} \)
73 \( 1 - 1.25e8T + 5.88e16T^{2} \)
79 \( 1 - 1.68e8T + 1.19e17T^{2} \)
83 \( 1 - 2.60e8T + 1.86e17T^{2} \)
89 \( 1 - 6.26e8T + 3.50e17T^{2} \)
97 \( 1 + 3.81e8T + 7.60e17T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.95074633690262211124091745696, −9.959771851936681497257936318115, −8.793210419269309315579870814161, −7.88657814681465688660742343037, −7.49835338530664893668254435566, −5.32449178263742103857248685580, −4.53722901958951024230134487544, −3.54358410357225273106957077132, −2.20109354769548604789409875274, −0.35603105079820288194391628244, 0.35603105079820288194391628244, 2.20109354769548604789409875274, 3.54358410357225273106957077132, 4.53722901958951024230134487544, 5.32449178263742103857248685580, 7.49835338530664893668254435566, 7.88657814681465688660742343037, 8.793210419269309315579870814161, 9.959771851936681497257936318115, 10.95074633690262211124091745696

Graph of the $Z$-function along the critical line